2025/26 AHSMC Part 1 Problem 3

Problem 3 of 16

I have two six-sided dice: a white one and a red one. Suppose the white one is fair, but the red die is “loaded” such that rolling a “ $1$ ” is twice as likely as rolling any other number. What is the probability that I roll a sum of “ $7$ ” with these two dice?

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Solution

The loaded die has $P(1) = 2/7$ and $P(2)=\cdots=P(6)=1/7$ . The six ways to total $7$ contribute $\frac{1}{6}\cdot\frac{1}{7} + \frac{1}{6}\cdot\frac{1}{7} + \frac{1}{6}\cdot\frac{1}{7} + \frac{1}{6}\cdot\frac{1}{7} + \frac{1}{6}\cdot\frac{1}{7} + \frac{1}{6}\cdot\frac{2}{7} = \frac{1}{6}.$ Alternatively: whatever the loaded die shows, the fair die has probability $1/6$ of making the sum $7$ . The answer is (D).

University of Alberta

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